Plasma oscillation
Plasma oscillations, also known as Langmuir waves (after ), are rapid oscillations of the in conducting media such as or s in the region. The oscillations can be described as an instability in the . The frequency only depends weakly on the wavelength of the oscillation. The resulting from the of these oscillations is the . Langmuir waves were discovered by American and in the 1920s. They are parallel in form to waves, which are caused by gravitational instabilities in a static medium. Mechanism Consider an electrically neutral plasma in equilibrium, consisting of a gas of positively charged s and negatively charged . If one displaces by a tiny amount an electron or a group of electrons with respect to the ions, the pulls the electrons back, acting as a restoring force. 'Cold' electrons If the thermal motion of the electrons is ignored, it is possible to show that the charge density oscillates at the plasma frequency : \omega_{\mathrm{pe}} = \sqrt{\frac{n_\mathrm{e} e^{2}}{m^*\varepsilon_0}}, \left\mathrm{rad/s}\right ( ), : \omega_{\mathrm{pe}} = \sqrt{\frac{4 \pi n_\mathrm{e} e^{2}}{m^*}}, rad/s ( ), where '' n_\mathrm{e} '' is the of electrons, '' e '' is the , '' m^* '' is the of the electron, and \varepsilon_0 is the . Note that the above is derived under the that the ion mass is infinite. This is generally a good approximation, as the electrons are so much lighter than ions. (This expression must be modified in the case of electron- plasmas, often encountered in ). Since the is independent of the , these s have an and zero . Note that, when m^*=m_\mathrm{e} , the plasma frequency, \omega_{\mathrm{pe}} , depends only on s and electron density n_\mathrm{e} . The numeric expression for angular plasma frequency is : f_\text{pe} = \frac{\omega_\text{pe}}{2\pi}~\left\text{Hz}\right Metals are only transparent to light with a frequency higher than the metal's plasma frequency. For typical metals such as aluminium or silver, '' n_\mathrm{e} '' is approximately 1023 cm-3, which brings the plasma frequency into the ultraviolet region. This is why most metals reflect visible light and appear shiny. 'Warm' electrons When the effects of the thermal speed v_{\mathrm{e,th}} = \sqrt{\frac{k_\mathrm{B} T_{\mathrm{e}}}{m_\mathrm{e}}} are taken into account, the electron pressure acts as a restoring force as well as the electric field and the oscillations propagate with frequency and related by the longitudinal Langmuir wave: : \omega^2 =\omega_{\mathrm{pe}}^2 +\frac{3k_\mathrm{B}T_{\mathrm{e}}}{m_\mathrm{e}}k^2=\omega_{\mathrm{pe}}^2 + 3 k^2 v_{\mathrm{e,th}}^2 , called the - . If the spatial scale is large compared to the , the s are only weakly modified by the term, but at small scales the pressure term dominates and the waves become dispersionless with a speed of \sqrt{3} \cdot v_{\mathrm{e,th}} . For such waves, however, the electron thermal speed is comparable to the , i.e., : v \sim v_{\mathrm{ph}} \ \stackrel{\mathrm{def}}{=}\ \frac{\omega}{k}, so the plasma waves can electrons that are moving with speed nearly equal to the phase velocity of the wave. This process often leads to a form of collisionless damping, called . Consequently, the large-''k'' portion in the is difficult to observe and seldom of consequence. In a plasma, fringing electric fields can result in propagation of plasma oscillations, even when the electrons are cold. In a or , the effect of the s' periodic potential must be taken into account. This is usually done by using the electrons' in place of m. References Category:Electricity